Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A

Average Error: 0.3 → 0.3

Time: 15.6s

Precision: binary64

Cost: 26368

Math

\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

↓

\[\left(x \cdot 0.5 - y\right) \cdot \left({\left(\sqrt{e^{t}}\right)}^{t} \cdot \sqrt{z \cdot 2}\right) \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))

↓

(FPCore (x y z t)
 :precision binary64
 (* (- (* x 0.5) y) (* (pow (sqrt (exp t)) t) (sqrt (* z 2.0)))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
}

↓

double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * (pow(sqrt(exp(t)), t) * sqrt((z * 2.0)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * exp(((t * t) / 2.0d0))
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * 0.5d0) - y) * ((sqrt(exp(t)) ** t) * sqrt((z * 2.0d0)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.exp(((t * t) / 2.0));
}

↓

public static double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * (Math.pow(Math.sqrt(Math.exp(t)), t) * Math.sqrt((z * 2.0)));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.exp(((t * t) / 2.0))

↓

def code(x, y, z, t):
	return ((x * 0.5) - y) * (math.pow(math.sqrt(math.exp(t)), t) * math.sqrt((z * 2.0)))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * exp(Float64(Float64(t * t) / 2.0)))
end

↓

function code(x, y, z, t)
	return Float64(Float64(Float64(x * 0.5) - y) * Float64((sqrt(exp(t)) ^ t) * sqrt(Float64(z * 2.0))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
end

↓

function tmp = code(x, y, z, t)
	tmp = ((x * 0.5) - y) * ((sqrt(exp(t)) ^ t) * sqrt((z * 2.0)));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[(N[Power[N[Sqrt[N[Exp[t], $MachinePrecision]], $MachinePrecision], t], $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.3
Target	0.3
Herbie	0.3

\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{1}\right)}^{\left(\frac{t \cdot t}{2}\right)} \]

Derivation

1. Initial program 0.3

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

2. Simplified 0.3

   \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot {\left(\sqrt{e^{t}}\right)}^{t}\right)} \]
   Proof
   (*.f64 (-.f64 (*.f64 x 1/2) y) (*.f64 (sqrt.f64 (*.f64 z 2)) (pow.f64 (sqrt.f64 (exp.f64 t)) t))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (*.f64 x 1/2) y) (*.f64 (sqrt.f64 (*.f64 z 2)) (pow.f64 (Rewrite<= exp-sqrt_binary64 (exp.f64 (/.f64 t 2))) t))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (*.f64 x 1/2) y) (*.f64 (sqrt.f64 (*.f64 z 2)) (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 (/.f64 t 2) t))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (*.f64 x 1/2) y) (*.f64 (sqrt.f64 (*.f64 z 2)) (exp.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 t t) 2))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (*.f64 x 1/2) y) (sqrt.f64 (*.f64 z 2))) (exp.f64 (/.f64 (*.f64 t t) 2)))): 2 points increase in error, 1 points decrease in error

3. Final simplification 0.3

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left({\left(\sqrt{e^{t}}\right)}^{t} \cdot \sqrt{z \cdot 2}\right) \]

Alternatives

Alternative 1
Error	0.3
Cost	13632

\[\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z + z\right) \cdot e^{t \cdot t}} \]

Alternative 2
Error	31.4
Cost	7112

\[\begin{array}{l} t_1 := x \cdot \sqrt{0.5 \cdot z}\\ \mathbf{if}\;x \leq -1.0103295863534678 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.545635814003035 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	1.2
Cost	6976

\[\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} \]

Alternative 4
Error	31.8
Cost	6720

\[x \cdot \sqrt{0.5 \cdot z} \]

Reproduce

herbie shell --seed 2022294 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :herbie-target
  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (exp 1.0) (/ (* t t) 2.0)))

  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))